A350390 -id:A350390 - cp's OEIS frontend

A286324 a(n) is the number of bi-unitary divisors of n.

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 6, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8

Offset: 1

Michel Marcus, May 07 2017

a(n) is the number of terms of the n-th row of A222266.

			From _Michael De Vlieger_, May 07 2017: (Start)
a(1) = 1 since 1 is the empty product; all divisors of 1 (i.e., 1) have a greatest common unitary divisor that is 1. 1 is a unitary divisor of all numbers n.
a(p) = 2 since 1 and p have greatest common unitary divisor 1.
a(6) = 4 since the divisor pairs {1, 6} and {2, 3} have greatest common unitary divisor 1.
a(24) = 8 since {1, 24}, {2, 12}, {3, 8}, {4, 6} have greatest unitary divisors {1, {1, 3, 8, 24}}, {{1, 2}, {1, 3, 4, 12}}, {{1, 3}, {1, 8}}, {{1, 4}, {1, 2, 3, 6}}: 1 is the greatest common unitary divisor among all 4 pairs. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
D. Suryanarayana, The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions pp 273-282, Lecture Notes in Mathematics book series (LNM, volume 251).
Eric Weisstein's World of Mathematics, Biunitary Divisor.

Cf. A222266, A188999, A293185 (indices of records), A340232, A350390.

Cf. A000005, A034444 (unitary), A037445 (infinitary).

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, 1 &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 90}] (* Michael De Vlieger, May 07 2017 *)
f[p_, e_] := If[OddQ[e], e + 1, e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120] (* Amiram Eldar, Dec 19 2018 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = #biudivs(n);

a(n)={my(f=factor(n)[,2]); prod(i=1, #f, my(e=f[i]); e + e % 2)} \\ Andrew Howroyd, Aug 05 2018

for(n=1, 100, print1(direuler(p=2, n, (X^3 - X^2 + X + 1) / ((X-1)^2 * (X+1)))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = e + (e mod 2). - Andrew Howroyd, Aug 05 2018
a(A340232(n)) = 2*n. - Bernard Schott, Mar 12 2023
a(n) = A000005(A350390(n)) (the number of divisors of the largest exponentially odd number dividing n). - Amiram Eldar, Sep 01 2023
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Let f(s) = Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (p-1)/((p+1)*p^2)) = A306071 = 0.80733082163620503914865427993003113402584582508155664401800520770441381...,
f'(1) = f(1) * Sum_{p prime} 2*(p^2 - p - 1) * log(p) /(p^4 + 2*p^3 + 1) = f(1) * 0.40523703144422392508596509911218523410441417240419849262346362977537989... = f(1) * A306072
and gamma is the Euler-Mascheroni constant A001620. (End)

A350389 a(n) is the largest unitary divisor of n that is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Amiram Eldar, Dec 28 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001222, A007913, A268335, A350387, A350388, A350390.

f[p_, e_] := If[OddQ[e], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^f[i,2], 1));} \\ Amiram Eldar, Sep 18 2023

from math import prod
from sympy import factorint
def A350389(n): return prod(p**e if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022

Multiplicative with a(p^e) = p^e if e is odd and 1 otherwise.
a(n) = n/A350388(n).
A001222(a(n)) = A350387(n).
a(n) = 1 if and only if n is a positive square (A000290 \ {0}).
a(n) = n if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ (1/2)*c*n^2, where c = Product_{p prime} (1 - p/(1+p+p^2+p^3)) = 0.7406196365...
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1)). - Amiram Eldar, Sep 18 2023

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Jul 29 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064549, A007913, A268335, A336643, A350390.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356192, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^e, p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^f[i,2], f[i,1]^(f[i,2]+1)))};

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A268335.
a(n) = A064549(n)/A007913(n).
a(n) = n*A336643(n).
a(n) = n^2/A350390(n).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)

A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 07 2019

First differs from A050361 at n = 64.
From Amiram Eldar, Sep 08 2023: (Start)
The number of exponentially odd divisors of n is A322483(n), and their sum is A033634(n).
A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n. (End)
Also, the number of divisors of n that are cubefull exponentially odd numbers (A335988). - Amiram Eldar, Feb 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033634, A050361, A065487, A268335, A295316, A335988, A350390.

Cf. A003557, A005361 (number of coreful divisors), A046951, A268335.

fun[p_,e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); Array[a, 100]

a(n) = vecprod(apply(x -> (x+1)\2, factor(n)[, 2])); \\ Amiram Eldar, Sep 01 2023

Multiplicative with a(p^e) = floor((e+1)/2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Sep 10 2022
a(n) = A046951(A350390(n)) (the number of squares dividing the largest exponentially odd divisor of n). - Amiram Eldar, Sep 01 2023
From Amiram Eldar, Sep 08 2023: (Start)
a(n) = A046951(A003557(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)). (End)

Name corrected by Amiram Eldar, Sep 08 2023

A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 10 2022

			a(27) = 9 since 9 = 3^2 is the largest divisor of 27 with an exponent in its prime factorization, 2, that is a power of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A138302, A353898.

Similar sequences: A000265, A007947, A008834, A055071, A350390.

f[p_, e_] := p^(2^Floor[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = p^(2^floor(log_2(e))).
a(n) = n if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.4616988732... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k)) and f(0) = 0.

A102631 a(n) = n^2 / (squarefree kernel of n).

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 32, 27, 10, 11, 24, 13, 14, 15, 128, 17, 54, 19, 40, 21, 22, 23, 96, 125, 26, 243, 56, 29, 30, 31, 512, 33, 34, 35, 216, 37, 38, 39, 160, 41, 42, 43, 88, 135, 46, 47, 384, 343, 250, 51, 104, 53, 486, 55, 224, 57, 58, 59, 120, 61, 62, 189, 2048, 65, 66, 67

Offset: 1

Reinhard Zumkeller, Feb 25 2005

Index of first occurrence of n in A019554. - Franklin T. Adams-Watters, Nov 17 2006

Michel Marcus, Table of n, a(n) for n = 1..5000

Cf. A000290, A007947, A003557, A005117, A019554, A350390.

a[n_] := n^2/Times @@ FactorInteger[n][[All, 1]];
Array[a, 70] (* Jean-François Alcover, Jun 11 2019 *)

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,2] = 2*f[k,2]-1); factorback(f); \\ Michel Marcus, Aug 20 2017

def A102631(n) :
    p = n
    for a in factor(n) :
        if a[1] > 1 :
            p = p * a[0]^(a[1]-1)
    return p
[A102631(n) for n in (1..67)] # Peter Luschny, Feb 07 2012

a(n) = A000290(n)/A007947(n) = n*A003557(n);
a(n) = n iff n is squarefree: a(A005117(n)) = A005117(n).
Multiplicative with a(p^e) = p^{2e-1}. - Franklin T. Adams-Watters, Nov 17 2006
Dirichlet g.f.: Product_{p prime} (1 - p/(p^2 - p^s)). - Amiram Eldar, Aug 28 2023
a(n) = A350390(n^2). - Amiram Eldar, Nov 30 2023

A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 28 2020

a(n) is the least number k such that k*n (and also n/k) is an exponentially odd number (A268335). - Amiram Eldar, Nov 18 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A007913, A007947, A059841, A268335, A336644, A350390, A356191.

f[p_, e_] := p^(1 - Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336643(n) = (factorback(factorint(n)[, 1]) / core(n));

A336643(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(1-(f[i, 2]%2))));

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X^2) * (1 + X + p*X^2 - X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A336643(n): return prod(primefactors(n))//core(n) # Chai Wah Wu, Dec 30 2021

def A336643(n: int) -> int:
    return prod(b^(1 - e % 2) for (b, e) in list(factor(n)))
print([A336643(n) for n in range(1, 106)])  # Peter Luschny, Aug 23 2025

a(n) = A007947(n) / A007913(n).
Multiplicative with a(p^k) = p^(1-(k mod 2)) = p^A059841(k).
a(n) = n/A350390(n). - Amiram Eldar, Jan 01 2022
a(n) = A356191(n)/n. - Amiram Eldar, Nov 18 2022
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Sep 09 2023
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(1-5*s) + p^(2-5*s) + 2*p^(1-4*s) - p^(2-4*s) - p^(1-3*s) + p^(-3*s) - 2*p^(-2*s)).
Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 12 * (log(n) + 3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.217778716619536378323007514119446813130797755001355937648276403523626491...,
f'(1) = f(1) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.716596820856763078660955244870812634072512131626849517007098664560806248...
and gamma is the Euler-Mascheroni constant A001620. (End)

A372379 The largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 24, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70

Offset: 1

Amiram Eldar, Apr 29 2024

First differs from A350390 at n = 32.
The largest term of A036537 dividing n.
The largest divisor of n whose exponents in its prime factorization are all of the form 2^k-1 (A000225).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000225, A000523, A036537, A138302, A162643, A282940, A350390, A353897, A372380, A372381.

f[p_, e_] := p^(2^Floor[Log2[e + 1]] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2^exponent(f[i, 2]+1)-1));}

from math import prod
from sympy import factorint
def A372379(n): return prod(p**((1<<(e+1).bit_length()-1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Apr 30 2024

Multiplicative with a(p^e) = p^(2^floor(log_2(e+1)) - 1).
a(n) = n if and only if n is in A036537.
a(A162643(n)) = A282940(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.7907361848... = Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k))-1 for k >= 1, and f(0) = 0.

A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 25 2023

First differs from A252505 at n = 64.

The sum of these divisors is A365174(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A004524, A046101, A005117, A252505, A268335, A350390, A325837, A365171, A365174.

f[p_, e_] := Floor[(e + 5)/4] + Floor[(e + 6)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+5)\4 + (x+6)\4, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X^2 - X^4)/((1 - X)^2*(1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Multiplicative with a(p^e) = floor((e+5)/4) + floor((e+6)/4) = A004524(e+5).
a(n) <= A000005(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034444(n), with equality if and only if n is squarefree (A005117).
a(n) == 1 (mod 2) if and only if n is a square of an exponentially odd number (i.e., a number whose prime factorization include only exponents e such that e == 2 (mod 4)).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/(p^2*(1 + p^2))) = 0.937494282731300250789438325050116436995101826036120273493270589183132928...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 + 2) * log(p) / (p^6 + 2*p^4 - 1) = f(1) * 0.192452062257404507109731932640803706644036700262364333369815000973104583...
and gamma is the Euler-Mascheroni constant A001620. (End)

A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 64.

Differs from A363332 at n = 1, 216, 432, 648, 864, 1000, ... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A109613, A350390, A368711.
Cf. A331273, A363332.
Similar sequences: A007424, A368781, A372602, A372603, A372604.

f[n_] := n - If[EvenQ[n], 1, 0]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = (n+1) \ 2 * 2 - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A350390(n)).
a(n) = A109613(A051903(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{i>=1} (1 - (1/zeta(2*i+1))) = 1.42929441950714075659... .

cp's OEIS Frontend

A286324 a(n) is the number of bi-unitary divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A350389 a(n) is the largest unitary divisor of n that is an exponentially odd number (A268335).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A102631 a(n) = n^2 / (squarefree kernel of n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica